An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

Abstract

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

“…One of these directions is the study of symbolic approaches related to non-linear ODEs and systems, the so-called reaction-diffusion, cross-diffusion equations [13,14]. Further, the research group is ambitious in studying other types of non-linear ODEs, which are still intensively researched today [15][16][17]. The second research area is the inverse problems involving second-order differential equations, which are also non-linear mathematical problems [18].…”